Selling to Intermediaries: Auction Design in a Common Value Model

Transcription

1 Selling to Intermediaries: Auction Design in a Common Value Model Dirk Bergemann Yale December 2017 Benjamin Brooks Chicago Stephen Morris Princeton Industrial Organization Workshop 1

2 Selling to Intermediaries a single object is sold to one of N intermediaries winning intermediary oers the object to the downstream client with highest willingness to pay (at take-it-or-leave-it price) nal value is common among all intermediaries each intermediary receives private signal about nal value each signal is willingness to pay of a distinct client thus, intermediaries have pure common value: equal to the maximum of the individual signals what is the revenue maximizing selling mechanism? 2

3 Auction Design in Pure Common Value Model more broadly, private signal gives partially information about strength of secondary market key feature: conditional on highest signal, the other signals contain no additional information about the common value related interpretations where bidders are not intermediaries: 1. a pure common value model (as an alternative specation of the wallet game (Myerson)) 2. frictionless resale market with independent private value: interim owner can make take-it-or-leave-it price under complete information 3

4 Revenue Maximizing Design characterize revenue maximizing auction maximal revenue is obtained by strikingly simple mechanism, stated at interim level (given signal of bidder i) 1. constant signal independent participation fee 2. constant signal independent probability of getting object contrast with rst, second, optimal auction with private values 4

5 Revenue Maximizing Design: Posted Price optimal mechanism shares some features with posted price 1. constant signal independent price it exactly coincides with posted price if 2. constant signal independent probability is 1/N necessary and sucient condition when optimal mechanism reduces exactly to posted price if posted price is an optimal mechanism it is inclusive: every bidder with every signal realization is willing to buy 5

6 Revenue Maximizing Design: Beyond Posted Price in generally aggregate assignment probability may be < 1 interim probability of getting object is constant and < 1/N ex post probability fori depends on entire signal prole conditionally on allocating the object optimal mechanism: 1. favors bidders with lower signals 2. discriminates against bidder with highest signal winner's blessing rather than winner's curse 6

7 A First Argument if seller could sell to intermediary without private information (or act as intermediary herself), then she could extract all surplus (expectation of best client value) with private information there are two competing forces: 1. would like to sell to intermediary with good client/signal, since has high unconditional willingness to pay 2. would like to sell to intermediary with bad client/signal, since he has less (private) information about best client value compromise: sell with equal probability to all intermediaries independent of their clients' values 7

8 Contributions: Substantive setting where bidders with higher signals have more accurate information about common value; arises in market with intermediaries, and many other settings: auctions for resources, IPO's countervailing screening incentives, tension between selling to 1. bidder with higher expected value and 2. bidder with less private information optimal to screen less - with no screening in inclusive limit foundation for posted price mechanisms new auction format: guaranteed demand auction 8

9 Contributions: Methodological very few results extend characterization of optimal auctions beyond private value case we extend optimal auctions into interdependent values: 1. with private values, local incentive constraints are sucient to pin down optimal mechanism 2. with interdependent values, global constraints matter, new arguments are required 9

10 Plan of Talk model of pure common values revenue equivalence and failure of rst order approach upper bound on revenue optimality of posted price mechanism construction of optimal mechanism implementation as guaranteed demand auction relationship to private values, power of optimal auctions 10

11 Model

12 Pure Common Value Model N bidders for a single object bidder i receives a signal s i [s, s] R + absolutely continuous cumulative distribution F (s i ), f (s i ) value is maximum of N independent signals: v (s 1,..., s N ) = max {s 1,..., s N } signal distribution F (s i ) induces value distribution G N (v): G N (v) = (F (s)) N value is rst order statistic of N independent signals 11

13 Utility and Allocation bidder i is expected utility maximizer with quasilinear preferences, probability q i of receiving object and transfers t i : feasibility of auction u i (s, q i, t i ) = v (s) q i t i q i (s) 0, with N q i (s) 1 ex post transfer t i (s) of bidder i, interim expected transfer: t i (s i ) = t i (s i, s i ) f i (s i ) ds i, s i S N 1 where i=1 f i (s i ) = j i f (s j ) 12

14 Incentive Compatibility bidder i surplus when reporting s i while observing s i : ( u i si, s i ) ( ) ( ) q i s i, s i v (si, s i ) f i (s i ) ds i t i s i s i SN 1 indirect utility given truthtelling is: u i (s i ) u i (s i, s i ) 13

15 direct mechanism {q i, t i } N i=1 is incentive compatible (IC) if u i (s i ) u i ( si, s i ), for all i and si, s i S and individually rational (IR) if u i (s i ) 0, for all i and s i S 14

16 Bidder Surplus and Revenue ex-ante bidder surplus is U i = s i S u i (s i ) f (s i ) ds i revenue is expected sum of transfers: R = N i=1 s i S t i (s i ) f (s i ) ds i seller maximizes R over all IC and IR direct mechanisms probability q (v), q i (v) object is assigned (to bidder i) given v total surplus is TS = s v=s vq (v) g N (v) dv 14

17 Three Related Papers Bulow & Klemperer (AER 1996) study optimal auctions with interdependent values: 1. provide a revenue equivalence theorem (we use/adapt it) 2. optimal allocation is sensitive to marginal value of private information 3. solve for special case when local constraints are sucient crucially, interim monotonicity of the allocation is neither necessary nor sucient for incentive compatibility Bulow & Klemperer (Rand 2002) introduce maximum game as common value model for mineral rights v (s 1,..., s N ) = max {s 1,..., s N } observe that winner's curse is so strong that second price auction is revenue dominated by posted price 15

18 Revenue Three Related Papers in First Price Auctions with General Information Structures: Implications for Bidding and Revenue (ECTA, 2017), we x distribution of values G N (v) and ask how common prior distribution of signals aect surplus and welfare 2/3 v 9 G(v) = v 2 on [0; 1] and N = 2 1/3 0 Welfare set with common values Welfare with v = max s 0 1/3 2/3 Bidder surplus Figure 1: Information and Surplus Distribution 16

19 Utility Three Related Papers we nd F (s(v)) = (G(v)) 1/N True type: Indirect utility in the FPA Reported type 17

20 Bounds on Bidder Surplus and Revenue

21 Review: Revenue Equivalence with Private Value standard approach since Myerson (1981) uses the constraint that truthful reporting is a local maximizer for the bidder local IC depends on interim probability of getting object q i (s i ) q i (s i, s i ) f i (s i ) ds i s i with private values, logic of envelope theorem implies a small increase in signal (=value) has direct eect only: Lemma (Revenue Equivalence) u i (s i ) = q i (s i ) In any incentive compatible direct mechanism indirect utility is u i (s i ) = u i (s) + si x=s q i (x) dx. 18

22 Revenue Equivalence with Maximum Value local IC depends on interim probability of getting object if signal s i is informative about v(s) if signal of i is highest among all bidders: q i (s i ) I si max j i s j q i (s i, s i ) f i (s i ) ds i s i with maximum value, logic of envelope theorem implies a small increase in signal (=value) has direct eect only if s i is highest: Lemma (Revenue Equivalence) u i (s i ) = q i (s i ) In any incentive compatible direct mechanism indirect utility is u i (s i ) = u i (s) + si x=s q i (x) dx. 19

23 Failure of Suciency of Local Constraints standard argument would establish that local incentive constraints also support global incentive constraints consider policy that assigns object to bidder with lowest signal q min i (s) = I si <min j i s j, this would imply that q i (x) = 0 and thus by the previous lemma: u i (s i ) = u i (s) + si x=s q i (x) dx = u i (s) optimal auction would leave bidders with zero surplus, would support ecient allocation...but then consider deviation of s i = s there are additional global constraints on q i (x)! 20

24 Global Incentive Constraints which global constraints matter for optimal revenue? previous argument suggest (all) downward deviations seller wants to distorts allocation to lower signals thus possibly all deterministic downward misreports, and hence all random downward misreports 21

25 A Relaxed Problem consider a smaller, one-dimensional, family of constraints: instead of reporting signal s i, report a random signal s i < s i, drawn from truncated prior on support [s, s i ]: F ( s i ) /F (si ) misreporting a redrawn lower signal analyze a relaxed problem which consists of local and small class of global constraints use these constraints to derive: 1. an upper bound on seller revenue 2. a lower bound on bidder utility 22

26 A Lower Bound on Bidder Utility what are the gains from misreporting a redrawn lower signal? equilibrium surplus of a bidder with type x is u i (s i ) = si x=s q i (x) dx surplus from misreporting the redrawn lower signal 1 F (s i ) si x=s u i (s i, x) f (x) dx gains vary depending on realized misreport average gains across all misreports are easy to compute 23

27 Average Gains from Misreporting misreport is redrawn from prior, bidder i is equally likely to fall anywhere in distribution of signals, unconditional on misreport, ex-ante likelihood that i receives object and x is highest signals q i (x) g N (x) if highest report is less than s i, surplus that bidder i obtains from being allocated object is s i rather than x, so s i x is dierence between deviator and truthtelling surplus: 1 F (s i ) si x=s [(s i x) q i (x) g N (x) + u i (x) f (x)] dx thus the incentive constraint requires: u i (s i ) 1 F (s i ) si x=s [(s i x) q i (x) g N (x) + u i (x) f (x)] dx 24

28 Lower Bound As Equality notice that si u i (s i ) = u i (s) + q i (x) dx x=s and thus q i (x) induces q i (x), connects local and global constraints u i (s i ) 1 F (s i ) si x=s [(s i x) q i (x) g N (x) + u i (x) f (x)] dx it also has to hold as sum across i : u(s) 1 s [(s x) q (x) g N (x) + u (x) f (x)] dx F (s) x=s lowest solution u (s) exists and solves inequality as equality can be integrated by parts as ( s ) 1 F (x) U = u (s) f (s) ds = dx q (s) g N (s) ds F (x) x S s x=s 25

29 A Generalized Virtual Utility Formula we obtain our nal formula for revenue, which is R = TS U = ψ (v) q (v) g N (v) dv where s ψ (v) = v x=v v 1 F (x) dx, F (x) compare to virtual utility in private value environments: π (x) = x 1 F (x) f (x) 26

30 Upper Bound on Revenue generalized virtual utility: s ψ (x) = x y=x Theorem (Revenue Upper Bound) 1 F (y) dy, F (y) In any auction in which the probability of allocation is given by q, bidder surplus is bounded below by U and expected revenue is bounded above by R. bound is valid for any allocation policy q(v) 27

31 Trade-Os: Eciency vs. Information Rents bound is generated by incentive constraint: u(s) 1 F (s) s x=s and rewriting in terms of q(x) and q(x): s y=s [(s x) q (x) g N (x) + u (x) f (x)] dx (s x) q (x) f (x) dx s x=s q (x) F (x) dx by allocation that favors low-signal bidders by making q(s) as small as possible and q(s) as large as possible: increasing q (x) has two competing eects on revenue: 1. increases total surplus generated by auction, 2. generates additional information rents for types greater than x it increases value of misreporting a redrawn lower signal 28

32 Posted Prices As Optimal Mechanism

33 Posted Prices consider mechanisms where object is always allocated pure common values allocation is therefore socially ecient Theorem (Revenue Optimality among Ecient Mechanisms) Among all mechanisms that allocate the object with probability one, revenue is maximized by setting a posted price of p = s v=s vg N 1 (v) dv, (1) i.e., expected value of object conditional on having lowest signal s. posted price is inclusive: all types purchase at p all bidders equally likely to receive object: q i (v) = 1/N, i, v. optimal selling mechanism is attained with constant interim transfer t = t i (s i ) and probability q = q i (s i ) 29

34 Optimality of Posted Price next, optimality of posted price among all possibly inecient mechanisms Corollary (Revenue Optimality of Posted Prices) A posted price mechanism is optimal if and only if s ψ(s) = s s 1 F (x) dx 0. F (x) If a posted price p is optimal, then it is fully inclusive. 30

35 Optimal Mechanism in General

36 Optimal Mechanism in General construct an novel mechanism/game that attains the revenue bound for all distributions guaranteed demand auction (GDA) direct mechanism to implement the bound exists as well descending clock (probability) auction implements revenue bound as well 31

37 The Guaranteed Demand Game: Rules bidder i demands d i [ 0, d ], where d is a parameter of game: 0 d i d 1/N let i denote identity of bidder with the highest demand: d i = max{d 1,..., d N } : if d i > 0: 1. i is allocated object with probability d i 2. bidder j i receives the object with probability (1 d i ) / (N 1) > d i if d i = 0, then the seller keeps the object 32

38 The Guaranteed Demand Game: Properties importantly if upper bound on demand d is: d < 1/N then conditional on highest demand being positive, d i > 0 bidder j is more likely to get object if he doesn't have highest demand: q j > di > d j each bidder's probability of receiving object is always at least his demand 33

39 Equilibrium Strategy unique equilibrium has a monotone pure strategy: ( ) 1 N 1 G N(r) G σ (s i ) = N (s i ) if s i r; 0 if s i < r, (2) where threshold r solves: G N (r) = 1 Nd. (3) and demand at s satises: σ (s) = ( 1 ( 1 Nd )) /N = d denote resulting equilibrium utility: u i (s) 34

40 The Guaranteed Demand Auction turn guaranteed demand game into guaranteed demand auction (GDA) by adding entry fees f i each bidder's message consists of a pair: entry decision and demand if bidder i decides to enter, he pays f i to the seller, and object is allocated among bidders who enter guaranteed demand game Proposition (Equilibrium of Guaranteed Demand Auction) As long as f i u i (s) for all i, it is an equilibrium for all bidders to enter the GDA and make demands according to (2). In equilibrium, bidders are indierent between their equilibrium demands and all lower demands. 35

41 Optimality of Guaranteed Demand Auction consider the GDA where the threshold r has zero generalized virtual utility: ψ (r ) = r s 1 F (y) y=r F (y) dy = 0 choice of d d = 1 G N (r ) N ensures that bidder makes a positive demand i bidder has signal greater than r Theorem (Optimality of Guaranteed Demand Auction) Revenue is maximized with a guaranteed demand auction with maximum demand d = (1 G N (r )) /N and a symmetric entry fee which is equal to u i (s). 36

42 Utility Incentive Constraints in Optimal Auction in the optimal auction, each bidder is indierent between his equilibrium bid and any lower bid 0.08 Indirect utility in the optimal auction True type: Reported type Figure 3: Uniform Downward Incentive Constraints and Winner's Blessing 37

43 Utility Incentive Constraints in First Price Auction in the rst price auction, each bidder is indierent between his equilibrium bid and any higher bid True type: Indirect utility in the FPA Reported type Figure 4: Uniform Upward Incentive Constraints and Winner's Curse 38

44 Where Do We use Maximum Game? which properties of the maximum game do we use, and where? 1. exact description of local incentive constraint in terms of probability q i (v) 2. exact description of gains of global incentive constraint in terms of s x 3. symmetry among losing bidders 39

45 Resale Interpretation seller biases the allocation towards those bidders who are likely to want to resell the object in secondary market since they have less private information about the resale value than the seller would have to incentivize them to reveal q cannot be too low, however, or else bidders would want to deviate by misreporting redrawn lower signals constraint boils down to the requirement that q (x) cannot be smaller than the probability that the object is allocated conditional on the highest signal being less than x 40

46 Implications

47 Maximum Game Bulow and Klemperer (2002) dene maximum game and show in second price auction in equilibrium each bidder bids his signal s i v = max j {s 1,..., s j,..., s N } in equilibrium, bidder with highest signal wins the auction and pays second-highest signal in fact, it is optimal to bid any amount which is at least your signal, and, in particular, it is optimal to bid your signal by contrast, in optimal auction, each bidder is indierent between reporting his signal and reporting any lower signal 41

48 Comparison with IPV suppose now that the signals are the values, thus independent private value environment: s i = v i max j {s 1,..., s N } in second price auction bidding his signal remains optimal thus, in second price auction of pure common value environment, each bidder behaves as if his signal is his true private value rather than a signal, and in particular a lower bound on the pure common value observation can be generalized 42

49 Strategic Equivalence consider independent private value model: v i (s 1,..., s N ) = s i denote the set of bidders with high signals { } H (s) = i s i = max s j j direct mechanism {q i, t i } is conditionally ecient if (i) q i (s) > 0 if and only if s i H (s) and (ii) there exists a cuto r such that the object is allocated whenever max i s i > r. Proposition Suppose a direct mechanism {q i, t i } is incentive compatible and individually rational for the independent private value model in which v i (s) = s i and that the allocation is conditionally ecient. Then {q i, t i } is also incentive compatible and individually rational for the maximum common value model in which v i (s) = max j {s j }. 43

50 Auctions vs Optimal Mechanism Bulow and Klemperer (1996) establish the limited power of optimal mechanisms as opposed to standard auction formats revenue of optimal auction with N bidders is strictly dominated by standard absolute auction with N + 1 bidders current common value environment is an instance of general interdependent value setting with one exception virtual utility functionor marginal revenue functionis not monotone due to maximum operator in common value model 44

51 A Closer Look at the Virtual Utility non-monotonicity leads to an optimal mechanism with features distinct from standard rst or second price auction. it elicits information from bidder with highest signal but minimizes probability of assigning him the object subject to incentive constraint virtual utility of each bidder, π i (s i, s i ): max j {s j }, if s i max{s i }; π i (s i, s i ) = max{s j } 1 F i (s i ) f i (s i ), if s i > max{s i }. downward discontinuity in virtual utility indicates why seller wishes to minimize the probability of assigning the object to the bidder with the high signal 45

52 Revenue Comparison virtual utility of bidder i fails monotonicity assumption even when hazard rate of distribution function is increasing everywhere BK (1996) require monotonicity of virtual utility when establishing their main result that an absolute English auction with N + 1 bidders is more protable than any optimal mechanism with N bidders revenue ranking does not extend to current auction environment compare revenue from optimal auction with N bidders to absolute, English or second-price, auction with N + K bidders absolute as there is no reserve price imposed 46

53 Reversal in Revenue Comparison Proposition (Revenue Comparison) For every N 1 and every K 1, the revenue from an absolute second-price auction with N + K bidders is strictly dominated by the revenue of an optimal auction with N bidders. comparison of second order statistic of N + K i.i.d. signals and rst order statistic of N + K 1 i.i.d. signals second order statistic of N + K signals is revenue of absolute second-price auction with N + K bidders. by earlier Theorem, optimal mechanism (weakly) exceeds revenue from a posted price set equal to the maximum of N + K 1 signals. now, if instead of N + K bidders, the optimal auction only has N bidders, then it is as if only N independent and identical distributed signals are revealed to the N bidders 47

54 Utility The Power of Optimal Auctions compare the indirect utility in the rst price auction and the optimal auction FPA versus optimal interim utility FPA Optimal auction Reported type Figure 5: Indirect Utility of Bidder with s = 1/2 across two mechanisms 48

55 Auctions with Resale characterization of the optimal mechanism remains valid if we interpret the model as one where the object is initially sold optimally among N bidders with independent private values then oered for resale under complete information in contrast to previous work on auctions with resale, such as Gupta and le Brun (1999) and Haile (2003) we analyze optimal mechanism in primary market Carroll and Segal (2016) study robust resale mechanisms 49

56 Conclusion characterized novel revenue maximizing auctions for a class of common value models common value models with qualitative feature that values are more sensitive to private information of bidders with more optimistic beliefs second interpretation as auction with frictionless resale market characterizations of optimal revenue that exist in the literature depend on information rents being smaller for bidders who are more optimistic about value qualitative impact is that earlier results found that optimal auctions discriminate in favor of more optimistic bidders today: optimal auctions discriminate in favor of less optimistic bidders since they obtain less information rents from being allocated the object 50

57 Appendix: Additional Slides

58 Optimal Auction construct an incentive compatible mechanism that exactly achieves the upper bound in the direct mechanism, all types are asked to make a xed payment, a participation fee, that is independent of their type no transfers beyond the participation fee are collected every type has the same interim expected probability of being allocated the object these two features of the optimal mechanism resemble a posted price mechanism unlike posted prices, the object is only allocated if the highest realized signal among the bidders exceeds a threshold value thus, typically, the probability that the object is assigned to some bidder is strictly smaller than one second feature distinct from posted prices is that the optimal mechanism discriminates against bidders with higher signals 51

59 Uniform Distribution family of translated uniform distributions on [a, a + 1], a > 0. marginal revenue function for these distributions is ψ a (x) = x lowest marginal revenue is ψ a (a) = a a+1 y=x a+1 y=a = a 1 N 1. thus posted price is optimal if ((x a) 1 N 1 ) dx, a > 1/ (N 1) ((x a) 1 N 1 ) 52